88 5. Divide 6. Divide a+2x by 4 a. 7. A man divided a dollar equally between 2 persons. What was the share of each? ANS. of a dollar. Each person has of the sum given; and of is. Here, the denominator is multiplied by the integer. In general, to divide a fraction by an integral quantity, we divide the numerator, when we can do so without a remainder; and when we cannot, we multiply the denominator. As these fractions have the same denominator, we divide the numerator of the dividend by the numerator of the divisor. It is clear that contains 3 times. α 15. Divide by с ad ANS. bs. We first reduce the fractions to a common denom с bc inator, and we find that, and =; and b d we then divide the numerator of the one by the numerator of the other, which gives us ad. But if we bc multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor, we shall obtain the same result: thus, x a X с = ad. Hence, to divide one fraction by another, we invert the divisor, and proceed as in multiplication. 90 FIRST LESSONS IN ALGEBRA. CHAPTER VII. POWERS. SECTION I. Involution of Simple Quantities. 1. WHAT is the product of a a multiplied by a? ANS. a3. The answer to this question, obtained by the common process of multiplication, is a a a. Instead of repeating the letter, we indicate the number of times it occurs in the product, by the small figure placed above it on the right hand. This figure is called its Exponent. It shows the Power of the letter; that is, how many times it is used as a factor in the multiplication. When a letter has no number annexed, the exponent is always a unit, or 1, and the letter is said to be the first power or Root; thus, a xa xa xaa is the fourth power, &c. The second power is sometimes called the square; the third power, the cube; and the fourth power, the biquadrate. If we suppose the value of a to be 3, a = 3, the first power; a2 = 32, or 3 × 3 = a3 33, or 3 x 3 x3= a1=34, or 3 X 3x3x3 = 9, the second power; 27, the third power; 81, the fourth power, &c. exponent of a quantity, be The coefficient and the ing very different things, must not be confounded together. Thus, the values of 5 x and x3 are far from being equal. Let the value of x be 6; then 5x5 X 6 = 30; but x5 = 6 × 6 × 6 × 6 × 6 = 7776. It is evident that we raise a single letter to any prɔposed power by giving it the exponent of that power. 2. What is the sixth power of x? the fourth power of d? the fifth power of c? the eighth power of a? the seventh power of x? 3. What is the product of a b multiplied by a b ; that is, what is the second power of a b? a b × ab=a a b b, or a2 b2. Here, the quantity to be raised consists of two factors, a and b; and the required power is expressed by the same factors, with the exponent of that power written above each. 4. What is the fourth power of x y? 5. Raise a b c to the sixth power. 6. What is the ninth power of a b x? 7. Involve m ny to the seventh power. 8. What is the square of 4 a b? ANS. 16 a2 b2. It will be remembered, that the square, or second power of any quantity, is the product of that quantity 92 FIRST LESSONS IN ALGEBRA. multiplied by itself; consequently, the second power of 4 a b is 4 ab × 4 a b = 16 a a b b, or 16 a2 b2. Hence, coefficients must be raised to any required power, by actual multiplication. 9. What is the third power of 5 a x? 10. What is the fourth power of 7 a b c? 11. Raise 6 x y z to the fifth power. 12. What is the fourth power of 2 a b c? 2 a b C, first power. +16 a4 b4c4, fourth power. Hence it appears, that when the root or first power is a negative quantity, the ODD powers are negative, and the EVEN powers are positive. 13. What is the second power of a3? ANS. a6. Here, the quantity to be involved is already a power, and the exponent is multiplied by the exponent of the power proposed; thus, a3×2 = a; for a = a a a, and a a a × a a a = ao. 14. What is the third power of a b3? ANS. a b9. 15. What is the second power of a3 b x2? 17. What is the fourth power of m4 x2 y? |